White Dwarf Stars

Introduction

A star produces light and heat by a process of nuclear fusion; lighter elements are fused together to form heavier elements at the same time liberating large quantities of energy. However this process cannot continue indefinitely - a net input of energy is required to form elements heavier than iron. Thus when a substantial fraction of a star's fuel has become converted to iron the star starts to collapse under its own gravity. If the star is not too heavy it may collapse to form a white dwarf star - a star in which individual atoms are compressed so closely together that electrons may move freely from one atom to another. The pressure to resist further gravitational collapse is then supplied by the electrons - it is termed degeneracy pressure. This is a quantum mechanical effect which prohibits electrons (indeed all spin-1/2 particles) from existing in the same quantum state. A force is then effectively generated when a sea of electrons, such as exist in the white dwarf star, is confined to a small volume.

We can analyze this situation within the context of Newtonian gravity by supposing that the gravitational force is balanced by a pressure due to the electron sea.

 

The righthand side is the attractive gravitational force on a spherical shell at radius r and this is then equated to the pressure gradient resisting compression. In addition we have the obvious equation relating the mass enclosed at radius r and the density .

 

Re-expressing the pressure P as a function of the density we can write eqn. 1 as

 

Eqns. 2 and 3 are two coupled first order differential equations for the mass and density profiles as a function of the radial coordinate r. These may be integrated (numerically) by assuming as initial conditions

The radius of the star will be the radius R at which the density . The mass of the star will then be M=m(R). What still needs to be input is an equation of state . To obtain this we assume that the electrons form a noninteracting Fermi gas. The (degeneracy) pressure can then be obtained by rather straightforward arguments. We will not detail these here but merely quote the result

The argument of the (known) function is simply related to the density where . The quantities d and c are pure numbers related to the electron and proton masses. Y is a dimensionless number that gives the number of electrons per nucleon. Different stars quit nuclear burning at slightly different stages so that this parameter will serve to distinguish classes of white dwarf star.

At this stage it is convenient to rescale the mass, density and radius variables into corresponding dimensionless ones

Rewriting eqns.2 and 3 in terms of these new variables results in the equations

• What the quantities , in terms of d,c ?
• How do they depend on Y?

For Y=1 , and .

• What are typical values for the radius, mass and density of our Sun ?
• Assume that . Show that in this case the equations can be rewritten as a single second order equation (the Lane-Emden equation)

   

• Write a code to integrate these equations for arbitrary Q(x) outward from some central density using a fourth order Runge Kutta scheme.
• Show that the solution to the Lane-Emden equation can be written as

  

  and

  

• Check your numerical code by comparing with the exact solution at these values.
• The correct form for Q(x) is given by

  

  Use your code to solve for the density and mass profile for central densities varying from 0.1 to . Draw a plot of how the mass and radius of the star vary with .

• You should see that as the mass approaches a limit - the Chandrasekhar limit and the radius becomes very small. No white dwarf star can be more massive than this. Stars that start out with more mass than this limit cannot stop gravitational collapse by using electron degeneracy pressure - they will continue to collapse to even greater densities - until the electrons combine with the protons to create neutrons. A neutron star is produced. Providing the initial mass is not too high this neutron gas can then halt the gravitational collapse using now neutron degeneracy pressure (which because of the larger mass of a neutron to an electron is significantly larger than the electron pressure).

  The calculation of the mass and density profiles is somewhat analogous to that presented here for the white dwarf. Again there will be an upper limit to the mass of the neutron star that can support itself against further gravitational collapse. If a star starts out above this limit and collapses without losing enough matter to put it below this limit no known physical mechanism can prevent collapse to a black hole.